The structure of infinite dimensional linear groups satisfying certain finiteness conditions
We review some recent results on the structure of infinite dimensional linear groups satisfying some finiteness conditions on certain families of subgroups. This direction of research is due to Leonid A. Kurdachenko, who developed the main steps of the theory jointly with mathematicians from several...
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| Zitieren: | The structure of infinite dimensional linear groups satisfying certain finiteness conditions / Jose M. Munoz-Escolano, Javier Otal, N.N. Semko // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 120–134. — Бібліогр.: 37 назв. — англ. |
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| citation_txt | The structure of infinite dimensional linear groups satisfying certain finiteness conditions / Jose M. Munoz-Escolano, Javier Otal, N.N. Semko // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 120–134. — Бібліогр.: 37 назв. — англ. |
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| description | We review some recent results on the structure of infinite dimensional linear groups satisfying some finiteness conditions on certain families of subgroups. This direction of research is due to Leonid A. Kurdachenko, who developed the main steps of the theory jointly with mathematicians from several countries.
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Algebra and Discrete Mathematics SURVEY ARTICLE
Number 4. (2009). pp. 120 – 134
c⃝ Journal “Algebra and Discrete Mathematics”
The structure of infinite dimensional linear
groups satisfying certain finiteness conditions
José M. Muñoz-Escolano, Javier Otal
and Nikolaj N. Semko
Communicated by I. Ya. Subbotin
Dedicated to Leonid with all our affection in his 60tℎ birthday
Abstract. We review some recent results on the structure of
infinite dimensional linear groups satisfying some finiteness condi-
tions on certain families of subgroups. This direction of research is
due to Leonid A. Kurdachenko, who developed the main steps of
the theory jointly with mathematicians from several countries.
Introduction
Let V be a vector space over a field F . We recall that a linear group G
is a subgroup of the group GL(V, F ) of all F–automorphisms of V under
composition of maps. The theory of linear groups is one of the topics
which has played a very important role in algebra and other branches
of mathematics. This importance occurs because of the well-known iso-
morphism between the group of all invertible n×n matrices with entries
in F , n ∈ ℕ, and linear groups, when n = dimFV (finite dimensional
linear groups), which gives rise to a rich interplay between geometrical
and algebraic ideas associated with such groups. Indeed the theory is
rich in many interesting and important results (see, for example, [5] and
[36], though the bibliography on the topic is very wide).
Supported by Proyecto MTM2007-60994 of Dirección General de Investigación del
Ministerio de Educación (Spain).
2000 Mathematics Subject Classification: 20F22, 20H20.
Key words and phrases: Linear group. Minimal, maximal, weak minimal and
weak maximal conditions. Central and augmentation dimensions.
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.J. M. Muñoz-Escolano, J. Otal, N. N. Semko 121
The study of the subgroups of GL(V, F ) in the case when V is infinite
dimensional over F has been much more limited and normally requires
some additional restrictions. The circumstances here are similar to those
present in the early development of Infinite Group Theory. One approach
there consisted in the application of finiteness conditions to the study of
infinite groups. One such restriction that has enjoyed considerable atten-
tion in linear groups is the notion of a finitary linear group. In the late
1980’s, R.E. Phillips, J.I. Hall and others studied infinite dimensional lin-
ear groups under finiteness conditions, namely finitary linear groups (see
[34, 14, 32, 35, 15, 16])). Here G is called finitary if, for each element
g ∈ G, the subspace CV (g) has finite codimension in V ; the reader is
referred to the above papers to see the type of results that have been ob-
tained. This is a good example of the effectiveness of finiteness conditions
in the study of infinite dimensional linear groups. Actually, a finitary lin-
ear group can be viewed as the linear analogue of an FC–group (group
with finite conjugacy classes); this association suggested that it was rea-
sonable to start a systematic investigation of these "infinite dimensional
linear groups" analogous to the fruitful study of finiteness conditions in
infinite group theory.
If G is a subgroup of GL(V, F ), then it is clear that G acts triv-
ially pointwise on the subspace CV (G), and hence G properly acts on
the factor-space V/CV (G). Leonid Kurdachenko realized this and, trying
to extend the concept of a finitary linear group, defined the central di-
mension of G as the F–dimension of V/CV (G) (denoted by centdimFG).
Thus G is finitary if and only if centdimF ⟨g⟩ is finite for every g ∈ G. At
this point, it is worth remarking that the above notion of central dimen-
sion of a linear group does not define a class of groups, since it heavily
relies in the way in which the linear group G is embedded in a particular
general linear group. In fact, given an abstract group G, it is easy to
construct embeddings of G in the same general linear group such that G
has infinite or finite central dimension depending on the embedding.
Example 1. Given a prime p, let A = ⟨a1⟩ × ⟨a2⟩ × . . . and B =
⟨b1⟩ × ⟨b2⟩ × . . . be two dos copies of the elementary abelian p–group.
Then B acts as A as follows:
⎧
⎨
⎩
a1 ⋅ bj = a1 j ≥ 1
aj+1 ⋅ bj = aj+1a1 j + 1 ≥ 2
ak ⋅ bj = ak k ∕= j + 1
We think of A as a vector space V over the prime field Fp of p elements
and of B as a subgroup G of the general linear group GL(V, F ). We have
that CV (G) = ⟨a1⟩ and hence G has infinite central dimension.
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.122 Infinite dimensional linear groups
Example 2. As above, let A = ⟨a1⟩×⟨a2⟩× . . . and B = ⟨b1⟩×⟨b2⟩× . . .
be copies of the elementary abelian p–group. In this case B acts as A as
follows:
{
a1 ⋅ bj = a1aj+1 j ≥ 1
ak ⋅ bj = ak k ≥ 2, j ≥ 1
As above, we think of A as a vector space V on Fp and of B as a subgroup
G of GL(V, F ). Since CV (G) = ⟨a2⟩ × ⟨a3⟩ . . . , G has finite central
dimension.
Consequently, we may not speak of the class of groups of finite central
dimension, and so we are focusing on specific and particular linear groups
that have an structure as abstract groups and not conversely. Even more,
the effect of the embedding can be very different of the idea we can
intuitively think. For example, a finite (linear) group can have infinite
central dimension as the following example shows.
Example 3. Let A = ⟨a1⟩ × ⟨a2⟩ × ⟨a3⟩ × . . . be an elementary abelian
p–group, p a prime and G = ⟨g⟩ be a cyclic group of order 2. We define
an action of G on A as follows:
{
a2j ⋅ g = a2j−1 j ≥ 1
a2j−1 ⋅ g = a2j j ≥ 1
Think of A as a vector space V over the field Fp of p elements and of G as
a subgroup of GL(V, Fp). In this example CV (G) = ⟨a1a2⟩× ⟨a3a4⟩× . . .
and so G has infinite central dimension.
Suppose that G has finite central dimension. If C = CG(CV (G)),
then C is a normal subgroup of G and G/C is isomorphic to a subgroup
of GL(n, F ), where n = centdimFG. Since C stabilizes the series
{0} ≤ CV (G) ≤ V,
we have that C is abelian. Moreover, if the characteristic of F is zero,
then C is a torsion-free abelian group, if the characteristic of F is the
prime p, then C is an elementary abelian p–group (see [13, Corollary
to Theorem 3.8] and [12, section 43] for these assertions). Therefore,
the structure of the given group G can be determined by the structure
of its factor-group G/C, which is an ordinary finite dimensional linear
group. In other words, finite dimensional central linear groups behave as
finite dimensional linear groups (or, more precisely, their structure can
be deduced from that ordinary case), so the efforts must be addressed to
infinite dimensional central linear groups.
In the recent years, Leonid Kurdachenko gave rise to a fundamental
idea to study these groups and started an intensive research following it.
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.J. M. Muñoz-Escolano, J. Otal, N. N. Semko 123
Let ℒicd(G) be the set of all proper subgroups of G of infinite central
dimension. In order to study infinite dimensional linear groups G that
are close to finite dimensional, Kurdachenko suggested to start making
ℒicd(G) very small in some sense. That is, by imposing some restrictions
to ℒicd(G). Given the big success that the study of infinite groups with
finiteness conditions had enjoyed, it seemed reasonable to study linear
groups with finiteness conditions in a similar way as the above.
In this paper we shall discuss some alternative approaches to the
study of a infinite dimensional central linear groups, based on the use of
different finiteness conditions, namely the different chain conditions. The
results quoted here present a survey of recent results in this direction of
research. As we mentioned above, this research was developed by Leonid
Kurdachenko and other mathematicians (as the authors of the present
work). We should mention that the next results are only a part of the
work of Kurdachenko in this topic. Actually, he was able to establish
the structure of infinite dimensional linear groups under other finiteness
conditions. For example, in [8, 3, 4, 2], an study of linear groups such
that every ℒicd(G)–subgroup has a finite (types of) ranks can be found,
while the description of linear groups with boundedly finite G–orbits or
related conditions can be seen in [10, 11, 9].
1. Minimal and maximal conditions on subgroups of infi-
nite central dimension
In the theory of infinite groups with finiteness conditions the first im-
portant problems were concerned with the maximal and minimal con-
ditions, which were first studied in ring and module theory. Soluble
groups with the maximal condition were considered by K.A. Hirsch, and
S.N. Cernikov began the investigation of groups with the minimal condi-
tion. Connected with these problems was the celebrated problem of O.Yu.
Schmidt concerning groups all of whose proper subgroups are finite. The
investigations which resulted from these problems determined the further
development of the theory of groups with finiteness conditions. If G is a
group and ℳ is a family of subgroups of G, we say that ℳ satisfies the
minimal (respectively the maximal) condition if given a descending chain
(respectively an ascending chain) {Hn ∣ n ∈ ℕ} of subgroups belonging
to ℳ there exists some m ∈ ℕ such that Hn = Hn+1 for all n ≥ m. The
former problems deal with the case when the family ℳ is composed by
all subgroups of G (G satisfies Min or G satisfies Max, respectively).
Let G ≤ GL(V, F ) be a linear group. We say that G satisfies the min-
imal condition for subgroups of infinite central dimension, or G satisfies
Min–icd, if the set ℒicd(G) satisfies the minimal condition. Similarly, we
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.124 Infinite dimensional linear groups
say that G satisfies the maximal condition for subgroups of infinite central
dimension, or G satisfies Max–icd, if the set ℒicd(G) satisfies the maxi-
mal condition. Linear groups satisfying Min–icd were studied by Dixon,
Evans and Kurdachenko in [6], while Kurdachenko and Subbotin gave a
complete description of the structure of linear groups satisfying Max–icd
in [30]. Both conditions above are trivially satisfied if ℒicd(G) = ∅, the
analogous problem for linear groups of infinite central dimension to O.Yu.
Schmidt’s foundational problem of Infinite Group Theory.
Theorem 1 ([6]). Suppose that G ≤ GL(V, F ) is a (locally soluble)–by–
finite subgroup of infinite central dimension whose proper subgroups have
finite central dimension. Then G ∼= Cp∞ , where p ∕=char F .
The next results connect linear groups satisfying Min–icd or Max–icd,
finitary linear groups and soluble groups.
Proposition 1 ([6, 30]). Suppose G ≤ GL(V, F ) and centdimFG infinite.
∙ If G satisfies Min–icd, then G is finitary or G satisfies Min.
∙ If G satisfies Max–icd, then G is finitary or G is finitely generated.
Proposition 2 ([6]). Let G be a subgroup of GL(V, F ) satisfying Min–
icd,
∙ If G is locally soluble, then G is soluble.
∙ If G is locally finite, then G is soluble–by–finite.
The structure of virtually soluble linear groups with Min–icd is the
main result of [6] (and extends a finite dimensional result of Malcev).
Theorem 2. [6] Let G ≤ GL(V, F ) be a (locally soluble)–by–finite sub-
group. Suppose that G has infinite central dimension and satisfies Min–
icd. If G is not Chernikov, then char F = p > 0 and G has a normal
series H ≤ D ≤ G such that:
(a) G/D is finite and D = H ⋋ Q where Q is a non-trivial divisible
Chernikov p′–subgroup of infinite central dimension;
(b) H is a nilpotent bounded p–subgroup of finite central dimension sat-
isfying the minimal condition on Q–invariant subgroups.
In particular, G is nilpotent–by–abelian–by–finite satisfying Min–n, the
minimal condition on normal subgroups.
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.J. M. Muñoz-Escolano, J. Otal, N. N. Semko 125
The description of linear groups with Max–icd is more complicated
than the above. We mention here a weak version of the structure theo-
rems, and refer to [30] for a detailed version.
The case when G is infinitely generated (G not finitely generated) is
studied first. In this case, G is finitary by Proposition 1.
Theorem 3 ([30]). Suppose that G ≤ GL(V, F ) is soluble, has infinite
central dimension and satisfies Max–icd. If G/G′ is infinitely generated,
then G has a normal series 1 ≤ H ≤ N ≤ L ≤ G such that
(1) L has finite index and infinite central dimension;
(2) L/H is abelian such that N/H is finitely generated, centdimFN is
finite and L/N ∼= Cq∞ where q ∕= char F .
(3) H is torsion-free nilpotent when char F = 0 and H is nilpotent
bounded p–subgroup when char F = p > 0.
In particular, G is nilpotent–by–abelian–by–finite.
Theorem 4 ([30]). Suppose that G ≤ GL(V, F ) is soluble, has infinite
central dimension and satisfies Max–icd. If G is infinitely generated, then
G has a normal subgroup S of infinite central dimension such that G/S
is finitely generated abelian–by–finite and S/S′ is infinitely generated.
Note that Theorem 3 can be applied to find out the structure of S.
We need the following notion. Let G ≤ GL(V, F ). Then
FD(G) = {x ∈ G ∣ ⟨x⟩ has finite central dimension}
is a normal subgroup of G ([6]), called the finitary radical of G.
We quote now the results for finitely generated groups.
Theorem 5 ([30]). Suppose that G ≤ GL(V, F ) is finitely generated
soluble, has infinite central dimension and satisfies Max–icd. If the central
dimension of FD(G) is finite, then G has a nilpotent normal subgroup
U ≤ FD(G) of finite central dimension such that
(1) G/U is polycyclic;
(2) U is torsion-free when char F = 0; and U is a bounded p–group
when char F = p > 0.
(3) U satisfies Max–⟨g⟩ for every g ∈ G ∖ FD(G).
In particular, G is nilpotent–by–polycyclic.
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.126 Infinite dimensional linear groups
Theorem 6 ([30]). Suppose that G ≤ GL(V, F ) is finitely generated
soluble, has infinite central dimension and satisfies Max–icd. If the central
dimension of FD(G) is infinite, then G has a normal subgroup L ≤
FD(G) of infinite central dimension such that
(1) G/L is abelian–by–finite and L/L′ is infinitely generated;
(2) L satisfies Max–⟨g⟩ for every g ∈ G ∖ FD(G).
The complete structure of G is obtained applying Theorem 3 to L.
2. Weak minimal and weak maximal conditions on infinite
central dimensional subgroups
The weak minimal and weak maximal conditions, introduced simulta-
neously in 1968 by R. Baer [1] and D. I. Zaitsev [37], are the natural
generalization of the ordinary minimal and maximal conditions. Let G
be a group and let ℳ be a family of subgroups of G. We say that ℳ sat-
isfies the weak minimal (respectively the weak maximal) condition if given
a descending chain (respectively an ascending chain) {Hn ∣ n ∈ ℕ} of ℳ–
subgroups, there exists some m ∈ ℕ such that the indices ∣Hn : Hn+1∣
(respectively ∣Hn+1 : Hn∣) are finite for all n ≥ m. Groups with the
weak minimal and maximal conditions for several families of subgroups
have been studied in many instances. For example, the weak chain con-
ditions on normal subgroups ([19, 21, 22]), subnormal subgroups ([20]),
non-normal subgroups ([23]) and non-subnormal subgroups ([28, 29]). See
also [31, Section 5.1] and [17] to obtain more information on this topic.
We say that a group G ≤ GL(V, F ) satisfies the weak minimal (re-
spectively maximal) condition for subgroups of infinite central dimension
(or briefly Wmin–icd (respectively Wmax–icd)) if ℒicd(G) satisfies the
weak minimal (respectively maximal) condition. Periodic linear groups
satisfying the conditions Wmin–icd and Wmax–icd were characterized by
the authors of this paper in [33]. We first mention an extension of a
well-known Zassehhaus’ result of finite dimensional linear groups.
Proposition 3 ([33]). Let G be a periodic subgroup of GL(V, F ) satisfy-
ing either Wmin–icd or Wmax–icd. Then
∙ If G is locally finite, then G is finitary or G is Chernikov.
∙ If G is locally soluble, then G is soluble.
In particular, periodic non-Chernikov locally soluble linear groups satis-
fying Wmin–icd or Wmax–icd are soluble finitary linear groups.
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.J. M. Muñoz-Escolano, J. Otal, N. N. Semko 127
The main result of [33] is the following theorem.
Theorem 7 ([33]). Let G be a periodic locally soluble subgroup of GL(V, F )
of infinite central dimension satisfying either Wmin–icd or Wmax–icd.
The following hold:
(1) If char F = 0, then G is Chernikov; and
(2) If char F = p > 0, then either G is Chernikov or G has a series of
normal subgroups H ≤ D ≤ G satisfying the following conditions:
(2a) H is a nilpotent bounded p–subgroup whose central dimension
is finite;
(2b) D has finite index and it is a semidirect product D = H ⋋
Q, where Q is a Chernikov divisible p′–group whose central
dimension is infinite; and
(2c) if K is a Prüfer q–subgroup of Q and centdimFK is finite,
then H has a finite K–composition series.
From this and Theorem 2, we deduce the following consequences.
Corollary 1 ([33]). Let G ≤ GL(V, F ) be a periodic locally soluble linear
group of infinite central dimension. Then the following conditions are
equivalent.
(1) G satisfies Wmin–icd,
(2) G satisfies Wmax–icd,
(3) G satisfies Min–icd.
Moreover, when G is a periodic locally nilpotent group and one of these
conditions holds then G is Chernikov.
Corollary 2 ([33]). Let G ≤ GL(V, F ) be a periodic locally soluble lin-
ear group of infinite central dimension. If G satisfies Max–icd, then G
satisfies Min–icd.
For non-periodic groups the situation is more complicated. Locally
nilpotent linear groups satisfying Wmin–icd and Wmax–icd were studied
in [25] and [27]. The next result shows that for nilpotent linear groups
the weak chain conditions on subgroups of infinite central dimensional
and the weak chain conditions on subgroups are equivalent.
Theorem 8 ([25]). Let G be a nilpotent subgroup of GL(V, F ) of infinite
central dimension satisfying either Wmin–icd or Wmax–icd. Then G is
minimax.
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.128 Infinite dimensional linear groups
Other results of [25] were deduced in characteristic prime only. In the
next result, t(G) is the torsion subgroup of the locally nilpotent group G,
GF the finite residual of G (the intersection of all subgroups of G of finite
index), and GN the nilpotent residual of G (the intersection of all normal
subgroups H of G such that G/H is nilpotent).
Theorem 9 ([25]). Let G ≤ GL(V, F ) be a locally nilpotent linear group
of infinite central dimension satisfying either Wmin–icd or Wmax–icd.
Suppose that char F = p > 0. Then
∙ G/t(G) is minimax
∙ G/GF is nilpotent and minimax
∙ G/GN is minimax
In particular, if t(G) has infinite central dimension, then G is minimax.
If G satisfies the weak minimal condition something more can be said.
Theorem 10 ([27]). Let G be a subgroup of GL(V, F ) of infinite central
dimension satisfying Wmin–icd.
(i) If G is locally nilpotent, then G is either minimax or finitary.
(ii) If G is hypercentral and char F = p > 0, then G is minimax.
Similar results for the condition Wmax–icd are false. Indeed in [27,
Section 4], an example of a hypercentral linear group satisfying Wmax–
icd, which is neither minimax nor finitary was constructed.
In [26], soluble linear groups satisfying Wmin–icd or Wmax–icd were
studied. The main result of that paper shows that their structure is
rather like the structure of finite dimensional soluble groups. Recall that
x ∈ G ≤ GL(V, F ) is unipotent if there exists some n ∈ ℕ such that
V (x − 1)n = 0. A subgroup H of G is called unipotent if every element
of H is unipotent and boundedly unipotent if n is independent of x.
Theorem 11 ([26]). Let G be a soluble subgroup of GL(V, F ) of infinite
central dimension satisfying Wmin–icd or Wmax–icd. Then either G is
minimax or G satisfies the following conditions:
(i) G has a normal boundedly unipotent subgroup L of finite central
dimension such that G/L is minimax;
(ii) L is a torsion-free nilpotent subgroup when char F = 0;
(iii) L is a bounded nilpotent p–subgroup when char F = p > 0.
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.J. M. Muñoz-Escolano, J. Otal, N. N. Semko 129
As in the locally nilpotent case we have
Theorem 12 ([26]). If G ≤ GL(V, F ) is soluble, has infinite central
dimensional and satisfies Wmin–icd, then G is either minimax or finitary.
We mention that non-minimax soluble finitary linear groups satisfying
Min–icd (hence Wmin–icd) can be constructed ([6, Section 5]). Also, it
is easy to construct non-finitary minimax soluble linear groups. On the
other hand, analogous results to above for the weak maximal condition
are not true. In fact, the example mentioned above ([27, Section 4]) is a
metabelian linear group satisfying Wmax–icd which is neither minimax
nor finitary. Summing up, our next result states the equivalence of weak
chain conditions in linear groups as the last result of this section.
Corollary 3. Let G ≤ GL(V, F ). If centdimFG is infinite and G is
∙ either locally nilpotent non-finitary; or
∙ soluble non-finitary; or
∙ hypercentral and char F = p > 0.
Then the following statements are equivalent
∙ G satisfies Wmin–icd,
∙ G satisfies Wmin,
∙ G satisfies Wmax,
∙ G is minimax
Moreover, if G is nilpotent, then the conditions are also equivalent to the
weak maximal condition on subgroups of infinite central dimension.
3. Augmentation dimension: antifinitary linear groups
If G ≤ GL(V, F ), G acts trivially on V/V (!FG), and hence G properly
acts on the subspace V (!FG). As in [7], the augmentation dimension
of G is defined to be the F–dimension of V (!FG) and is denoted by
augdimFG. The study of finite augmentation dimensional linear groups
can be reduced to the study of finite dimensional linear groups, just as
it happens for finite central dimensional linear groups. Once more, this
concept was introduced by Kurdachenko, when considering linear groups
whose infinite augmentation dimensional subgroups satisfy the minimal
condition ([7]). Later on, linear groups with some rank restrictions on
the same subgroups were studied ([4, 2]).
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.130 Infinite dimensional linear groups
Though the concept of augmentation dimension resembles to be op-
posite of the concept of central dimension, there is an apparently strong
relationship between them. In fact dimFV/CV (g)) = dimFV (g − 1) for
g ∈ GL(V, F ). More generally, if G ≤ GL(V, F ) is finitely generated,
then it is easy to see that the finiteness of one of the dimensions implies
the finiteness of the other. However, in the general case, this does not
hold as the following specific examples show.
Example 4. Let V and G as in Example 1. Since CV (G) = ⟨a1⟩ =
V (!(FpG)), we have that G has infinite central dimension and finite
augmentation dimension.
Example 5. Now, let V and G as in Example 2. In this case CV (G) =
⟨a2⟩ × ⟨a3⟩× = V (!FpG), and so G has finite central dimension and
infinite augmentation dimension.
In [18], some conditions over G such that the finiteness of centdimFG
implies the finiteness of augdimFG are established. Given a prime p, it
is said that an abstract group G has finite factor p-rank r (respectively,
finite factor 0-rank r) if whenever U and V are normal subgroups of G
such that V/U is an abelian p-group (respectively, torsion-free abelian)
and H is an intermediate subgroup U ≤ H ≤ V such that H/U is finitely
generated, then the minimal number of elements required to generated
H/U is less or equal to r, and r is the least such integer with this property.
Theorem 13 ([18]). Let G be a locally soluble subgroup of GL(V, F )
such that char F = p ≥ 0. Suppose that G has finite factor p–rank. If
centdimFG is finite, then so is augdimFG.
Theorem 14 ([18]). Let G ≤ GL(V, F ) be periodic and let char F = p >
0. Suppose that G has an ascending series of normal subgroups such that
every factor is either a p–group or a p′–group, and there is an integer r
such that every finitely generated p–subgroup of G can be generated by r
elements. If centdimFG is finite, then so is augdimFG.
As we mentioned above, finitary linear groups can be defined as those
linear groups whose finitely generated subgroups have finite augmentation
dimension. Therefore the following linear groups are in the antipodes
of finitary linear groups. We say that a group G ≤ GL(V, F ) is an
antifinitary linear group if each proper infinitely generated subgroup of G
has finite augmentation dimension. These groups were carefully studied
in [24]. We remark that the study has to be focused on non-finitary
antifinitary linear groups because of the next result.
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.J. M. Muñoz-Escolano, J. Otal, N. N. Semko 131
Theorem 15 ([7]). Let G be a (locally soluble)–by–finite linear group
of infinite augmentation dimension such that every proper subgroup has
finite augmentation dimension. Then G ∼= Cp∞ where p ∕=char F .
In particular, this happens if G is (locally soluble)–by–finite finitary
and antifinitary linear group of infinite central dimension. For ℒiac(G) =
∅ and it suffices to apply Theorem 15. Another interesting result about
locally finite antifinitary linear groups is the following
Proposition 4 ([24]). Let G ≤ GL(V, F ) be a locally finite antifinitary
linear group. If G is not finitary, then G is Chernikov.
The study of antifinitary linear groups splits in a natural way in two
cases depending on whether or not the group considered is finitely gener-
ated. We recall that a group G is said to be generalized radical if G has
an ascending series whose factors are locally nilpotent or locally finite.
For infinitely generated groups, we were able to establish that
Proposition 5 ([24]). Let G ≤ GL(V, F ) be an infinitely generated locally
generalized radical antifinitary linear group. If G is not finitary, then G
is locally finite.
Thus, Proposition 4 and Proposition 5 at once give that an infinitely
generated antifinitary linear group G that is locally generalized radical is
Chernikov. Even more, we have the following detailed description.
Theorem 16 ([24]). Let G be an infinitely generated locally generalized
radical antifinitary subgroup of GL(V, F ) such that G ∕= FD(G).
(1) If G/FD(G) is infinitely generated, then G is a Prüfer p–group for
some prime p.
(2) If G/FD(G) is finitely generated, then G = K⟨g⟩ satisfying the
following conditions:
(2a) K is a divisible abelian Chernikov normal q–subgroup of G, for
some prime q
(2b) g is a p–element, where p is a prime such that p = ∣G/FD(G)∣;
(2c) K is G–divisibly irreducible, i.e. K has no proper G–invariant
subgroups;
(2d) if q = p, then K has finite special rank equal to pm−1(p − 1)
where pm = ∣⟨g⟩/C⟨g⟩(K)∣ ; and
(2e) if q ∕= p, then K has finite special rank o(q, pm) where as above
pm = ∣⟨g⟩/C⟨g⟩(K)∣ and o(q, pm) is the order of q modulo pm.
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.132 Infinite dimensional linear groups
On the other hand, finitely generated antifinitary linear groups can
be characterized as follows.
Theorem 17 ([24]). Let G be a finitely generated radical antifinitary
subgroup of GL(V, F ). If G has infinite augmentation dimension, then
the following conditions holds:
(1) augdimFFD(G) is finite;
(2) G has a normal subgroup U such that G/U is polycyclic;
(3) U is boundedly unipotent and, in particular, U is nilpotent;
(4) U is torsion-free if char F = 0 and is a bounded p–subgroup if char
F = p > 0; and
(5) if
⟨1⟩ = Z0 ≤ Z1 ≤ ⋅ ⋅ ⋅ ≤ Zm = U
is an upper central series of U , then Z1/Z0, ..., Zm/Zm−1 are
finitely generated ℤ⟨g⟩–modules for each element g ∈ G ∖ FD(G).
In particular U satisfies the maximal condition on ⟨g⟩–invariant
subgroups (Max–⟨g⟩) for every g ∈ G ∖ FD(G).
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Contact information
J. M. Muñoz-
Escolano
Departamento de Matemáticas – IUMA,
Universidad de Zaragoza, Pedro Cerbuna
12. 50009 Zaragoza, SPAIN.
E-Mail: jmescola@unizar.es
J. Otal Departamento de Matemáticas – IUMA,
Universidad de Zaragoza, Pedro Cerbuna
12. 50009 Zaragoza, SPAIN.
E-Mail: otal@unizar.es
N. N. Semko Department of Mathematics, National State
Tax Service Academy of Ukraine, 09200 Ir-
pen, UKRAINE.
E-Mail: n−semko@mail.ru
Received by the editors: 23.08.2009
and in final form 23.08.2009.
|
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| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1726-3255 |
| language | English |
| last_indexed | 2025-12-07T17:41:27Z |
| publishDate | 2009 |
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| spelling | Jose M. Munoz-Escolano Javier Otal Semko, N.N. 2019-06-15T17:03:20Z 2019-06-15T17:03:20Z 2009 The structure of infinite dimensional linear groups satisfying certain finiteness conditions / Jose M. Munoz-Escolano, Javier Otal, N.N. Semko // Algebra and Discrete Mathematics. — 2009. — Vol. 8, № 4. — С. 120–134. — Бібліогр.: 37 назв. — англ. 1726-3255 2000 Mathematics Subject Classification:20F22, 20H20. https://nasplib.isofts.kiev.ua/handle/123456789/154635 We review some recent results on the structure of infinite dimensional linear groups satisfying some finiteness conditions on certain families of subgroups. This direction of research is due to Leonid A. Kurdachenko, who developed the main steps of the theory jointly with mathematicians from several countries. Supported by Proyecto MTM2007-60994 of Direcci ́on General de Investigaci ́on delMinisterio de Educaci ́on (Spain). en Інститут прикладної математики і механіки НАН України Algebra and Discrete Mathematics The structure of infinite dimensional linear groups satisfying certain finiteness conditions Article published earlier |
| spellingShingle | The structure of infinite dimensional linear groups satisfying certain finiteness conditions Jose M. Munoz-Escolano Javier Otal Semko, N.N. |
| title | The structure of infinite dimensional linear groups satisfying certain finiteness conditions |
| title_full | The structure of infinite dimensional linear groups satisfying certain finiteness conditions |
| title_fullStr | The structure of infinite dimensional linear groups satisfying certain finiteness conditions |
| title_full_unstemmed | The structure of infinite dimensional linear groups satisfying certain finiteness conditions |
| title_short | The structure of infinite dimensional linear groups satisfying certain finiteness conditions |
| title_sort | structure of infinite dimensional linear groups satisfying certain finiteness conditions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/154635 |
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